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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.oct_overview"></a><a class="link" href="oct_overview.html" title="Overview">Overview</a>
</h2></div></div></div>
<p>
      Octonions, like <a class="link" href="../quaternions.html" title="Chapter 17. Quaternions">quaternions</a>, are a relative
      of complex numbers.
    </p>
<p>
      Octonions see some use in theoretical physics.
    </p>
<p>
      In practical terms, an octonion is simply an octuple of real numbers (α,β,γ,δ,ε,ζ,η,θ), which
      we can write in the form <span class="emphasis"><em><code class="literal">o = α + βi + γj + δk + εe' + ζi' + ηj' + θk'</code></em></span>, where
      <span class="emphasis"><em><code class="literal">i</code></em></span>, <span class="emphasis"><em><code class="literal">j</code></em></span>
      and <span class="emphasis"><em><code class="literal">k</code></em></span> are the same objects as for quaternions,
      and <span class="emphasis"><em><code class="literal">e'</code></em></span>, <span class="emphasis"><em><code class="literal">i'</code></em></span>,
      <span class="emphasis"><em><code class="literal">j'</code></em></span> and <span class="emphasis"><em><code class="literal">k'</code></em></span>
      are distinct objects which play essentially the same kind of role as <span class="emphasis"><em><code class="literal">i</code></em></span>
      (or <span class="emphasis"><em><code class="literal">j</code></em></span> or <span class="emphasis"><em><code class="literal">k</code></em></span>).
    </p>
<p>
      Addition and a multiplication is defined on the set of octonions, which generalize
      their quaternionic counterparts. The main novelty this time is that <span class="bold"><strong>the multiplication is not only not commutative, is now not even
      associative</strong></span> (i.e. there are octonions <span class="emphasis"><em><code class="literal">x</code></em></span>,
      <span class="emphasis"><em><code class="literal">y</code></em></span> and <span class="emphasis"><em><code class="literal">z</code></em></span>
      such that <span class="emphasis"><em><code class="literal">x(yz) ≠ (xy)z</code></em></span>). A way of remembering
      things is by using the following multiplication table:
    </p>
<p>
      <span class="inlinemediaobject"><img src="../../octonion/graphics/octonion_blurb17.jpeg"></span>
    </p>
<p>
      Octonions (and their kin) are described in far more details in this other
      <a href="../../quaternion/TQE.pdf" target="_top">document</a> (with <a href="../../quaternion/TQE_EA.pdf" target="_top">errata
      and addenda</a>).
    </p>
<p>
      Some traditional constructs, such as the exponential, carry over without too
      much change into the realms of octonions, but other, such as taking a square
      root, do not (the fact that the exponential has a closed form is a result of
      the author, but the fact that the exponential exists at all for octonions is
      known since quite a long time ago).
    </p>
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<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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